1 Left Group
Calculus

Let the operation Eab correspond to
left division a-1b
in groups. Kalman [3]
shows that the following is an equational 2-basis for group
theory:

EExEyyEzz = x

EExyExz = Eyz

It is also shown in [3]
that the following is a 5-axiom logical calculus
(assuming condensed detachment as the sole rule of
inference) for groups (i.e., a =
b is a theorem of group theory iff both Eab
and Eba are derivable from the
following five axioms using only condensed detachment as the sole
rule of inference):

(L1) EEEpEEqqprr

(L2) EEEEEpqEprEqrss

(L3) EEEEEEpqEprsEEqrstt

(L4)
EEEEpqrsEEEptrEEqts

(L5) EEEpEEqprEEsptEEEEpqsrt

With OTTER, McCune [13]
showed (inter alia) that L1, L4,
L5 are dependent, and that the following 27-symbol
formula is a single axiom for LG:

EEEEpqrEEstEEEutEusvErEEqpv

Recently, we found the following 23-symbol axiom for LG (and
several others of the same length):

EEEpqrEEEspEEEtuEqsEutr

Open Question: Is there a shorter single axiom for LG?
The 11-symbol formula EEpqEEprEqr cannot be ruled-out using only
finite models.

2 Classical
Sentential Logic --- In the Sheffer Stroke (D)

In 1917, Nicod [21]
showed3 that
the following 23-symbol formula (in Polish notation) is a single
axiom for classical sentential logic (D is interpreted
semantically as NAND, i.e., the Sheffer stroke):

DDpDqrDDtDttDDsqDDpsDps

The only rule of inference for Nicod's single axiom system is
the following, somewhat odd, detachment rule for D:

From DpDqr and p, infer r.

Lukasiewicz [7]
later showed that the following substitution instance
(t/s) of Nicod's axiom (N) would suffice:

DDpDqrDDsDssDDsqDDpsDps

Lukasiewicz's student Mordchaj Wajsberg [37]
later discovered the following organic4
23-symbol single axiom for D:

DDpDqrDDDsrDDpsDpsDpDpq

Lukasiewicz later discovered another 23-symbol organic axiom:

DDpDqrDDpDrpDDsqDDpsDps2

We have discovered many new 23-symbol single axioms, some of
which are organic and have only 4 variables, e.g.,

DDpDqrDDpDqrDDsrDDrsDps

We can now report that the shortest single axioms for
these Sheffer Stroke systems contain 23 symbols. No shorter
axioms exist.

3 Classical
Sentential Logic --- In Implication (C) and The False (O)

Meredith [18,16]
reports two 19-symbol single axioms for classical sentential logic
(using only the rule of condensed detachment) in terms of
implication C and the constant O (semantically, O is ``The
False''):

CCCCCpqCrOstCCtpCrp
CCCpqCCOrsCCspCtCup

Meredith [18]
claims to have ``almost completed a proof that no single axiom of
(C,O) can contain less than 19 letters.'' As far as we know, no
such proof was ever completed (that is, until now...).

We have performed an exhaustive search/elimination of all
(C,O) theorems with fewer than 19 symbols. We have proven
Meredith's conjecture: no single axiom of (C,O) can contain
less than 19 letters.5

4 The
Implicational Fragment (C5) of the Modal Logic S5

In their classic paper [6],
Lemmon, Meredith, Meredith, Prior, and Thomas present several
axiomatizations (assuming only the rule of condensed detachment)
of the system C5, which is the strict implicational fragment of
the modal logic S5.

Bases for C5 containing 4, 3, 2, and a single axiom are
presented in [6]. The
following 2-basis is the shortest of these bases. It contains 20
symbols, 5-variables, and 9 occurrences of the connective C.

Cpp
CCCCpqrqCCqsCtCps

The following 21-symbol (6-variable, 10-C) single axiom (due
to C.A. Meredith) for C5 is also reported in [6]:

CCCCCttpqCrsCCspCuCrp

We searched both for new (hopefully, shorter than previously
known) single axioms for C5 and for new 2-bases for C5.

We discovered the following new 2-basis for C5, which (to the
best of our knowledge) is shorter than any previously known basis
(it has 18 symbols, 4 variables, and 8 occurrences of C):

Cpp
CCpqCCCCqrsrCpr

Moreover, we discovered the following new 21-symbol
(6-variable, 10-C) single axiom for C5 (as well as 5 others, not
given here):

CCCCpqrCCssqCCqtCuCpt

No formula with fewer than 21 symbols is a single axiom
for C5. And, no basis for C5
whatsoever has fewer than 18
symbols.

5 The
Implicational Fragment (C4) of the Modal Logic S4

C4 is the strict-implicational fragment of the modal logic S4
(and several other modal logics in the neighborhood of S4 - see
Ulrich's [34]).

As far as we know, the shortest known basis for C4 is due to
Ulrich (see Ulrich's [34]),
and is the following 25-symbol, 11-C, 3-axiom basis:

Cpp
CCpqCrCpq
CCpCqrCCpqCpr

Anderson & Belnap [1]
state the finding of a (short) single axiom for C4 as an open
problem (as far as we know, this has remained open). The
following is a 21-symbol (6-variable, 10-C) single axiom for
C4:

CCpCCqCrrCpsCCstCuCpt

We have also the following 20-symbol 2-basis for C4:

CpCqq
CCpCqrCCpqCsCpr

No formula with fewer than 21 symbols is a single axiom
for C4. And, no basis for C4
whatsoever has fewer than 20
symbols.

The ``classical'' relevance logic R-Mingle (RM) was first
carefully studied by Dunn in the late 60's (e.g., in
[2]). Interestingly, the
implicational fragment of R-Mingle (RM→)
has an older history.

RM→ was studied
(albeit, unwittingly!) by Sobocinski in the early 50's.
Sobocinski [30]
discusses a two-designated-value-variant of Lukasiewicz's
three-valued implication-negation logic (I'll call Sobocinski's
logic S). Sobocinski leaves the axiomatization of
S→ as an open
problem.

Rose [26,27]
solved Sobocinski's open problem, but his axiomatizations of
S→ are very
complicated and highly redundant (see Parks' [23]).

Meyer and Parks [19,24]
report an independent 4-axiom basis for S→.
They also show that S→
= RM→, thus providing
an independent 4-basis for RM→.
Meyer and Parks show that RM→
can be axiomatized by adding the following ``unintelligible''
21-symbol formula to R→:

CCCCCpqqprCCCCCqppqrr

In other words, the following is a 5-basis for RM→:

Cpp

CpCCpqq

CCpqCCrpCrq

CCpCpqCpq

CCCCCpqqprCCCCCqppqrr

The reflexivity axiom Cpp is dependent in the above 5-basis.
The remaining (independent) 4-basis is the Meyer-Parks basis for
RM→.

After much effort (and, with valuable assistance from Bob and
Larry), we discovered the following 13-symbol replacement for
Parks' 21-symbol formula (we've also shown that there are none
shorter):

The contraction axiom CCpCpqCpq is dependent in our new
4-basis. The remaining (independent) 3-basis for
RM→ contains 31
symbols and 14 C's (the Meyer-Parks basis has 4 axioms, 48
symbols, and 22 C's).

7 Single
Axioms for the Implicational Fragments of Some Other Non-Classical
Logics

It was shown by Rezus [25]
(building on earlier seminal work of Tarksi and Lukasiewicz
[8]) that the systems
E→, R→,
and L→have single
axioms. However, applying the methods of [25]
yields very long, inorganic single axioms. As far as we
know, these axioms have never been explicitly written down. Here
is a 69-symbol (17-variable!) single axiom for the implicational
fragment of Lukasiewicz's infinite-valued logic L→(obtained using the methods of [25]):

CCCfCgfCCCCCCCCCcdCCecCedCCaCbazzCCCCxyyCCyxxwwCCCCtuCutCutssCCqCrqpp

Single axioms of comparable length (i.e., containing
fewer than 75 symbols) can also be generated for the relevance
logics E→and
R→(omitted). Here's
what we know about the shortest single axioms for the systems
E→, R→,
L→, and RM→:

The shortest single axiom for E→has between 25 and 75 symbols.

The shortest single axiom for R→has between 25 and 75 symbols.

The shortest single axiom for L→has at most 69 symbols.

The shortest single axiom for RM→(if there is one7)
has at least 25 symbols.

8 Automated
Reasoning Techniques Used

First, we wrote computer programs to generate a large list of
candidate formulas which were to be tested as axioms. For most
problems, it was practical to generate an exhaustive list of all
formulas with up to twenty-one symbols (we can now do these
through 23 symbols).

All the formulas in the list would be tested (using matrices)
to see which are likely to be tautologies in the system in
question.8

We immediately eliminated large numbers of formulas by
applying known results about axiomatizations in the various
systems. For example, as reported by Lemmon et. al., every
axiomatization for C5 must contain a formula with Cpp as a
(possibly improper) subformula [6].
Another useful result for this purpose is the Diamond-McKinsey
theorem that no Boolean algebra can be axiomatized by formulas
containing less than three distinct propositional letters
[1].

A set of formulas was selected from the list at random. Using
either SEM [40] or a
program written by the authors, we found a matrix that respects
modus ponens, invalidates a known axiom-basis for the system, but
validates the formulas selected from the list. Such a model
suffices to show that the formulas are not single axioms for the
system.

All the remaining formulas in the list were then tested
against that matrix. Every formula validated by that matrix would
be eliminated.

Steps 4 and 5
were repeated until the list of candidate formulas was down to a
small number, or eliminated
entirely.

We then use OTTER [14]
(+strategies! - see [39]
and [4])
to attempt to prove a known axiom basis from each of the remaining
candidates.

3
Actually, Nicod's original proofs are erroneous (as noted by
Lukasiewicz in [7]). See
Scharle's [28] for a
rigorous proof of the completeness of Nicod's system.

4 A
single axiom is organic if it contains no tautologous
subformulae. Nicod's original single axiom (and Lukasiewicz's
simplifcation of it) are non-organic, because they contain
tautologous subformulae of the form DxDxx.

5 The
elimination of some (C,O) candidates relied on matrices generated
using local search techniques (as described by Ted Ulrich in
his [33, 35]).
Local search is very powerful in the context of implicational logics.
It has led to many useful models.

6 The
alternative 13-symbol formula CCCpCCCqprqrr will also serve this
purpose.

7
Rezus's work does not apply to RM→,
so whether RM→ has a
single axiom remains open.

8 We
say `likely to be tautologies' because C4 and C5 do not have finite
characteristic matrices. Thus, we used matrices which validate all
tautologies for the system, but also validate a small number of
contingent formulas.